Bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.
Preliminaries
Suppose that is a topological vector space with a continuous [dual space] and let for all andThe convex hull of a set denoted by is the smallest convex set containing
The convex balanced hull of a set is the smallest convex balanced set containing
The polar of a subset is defined to be:
while the prepolar of a subset is:
The bipolar of a subset often denoted by is the set
Statement in functional analysis
Let denote the weak topology on .Statement in convex analysis
Special case
A subset is a nonempty closed convex cone if and only if when where denotes the positive dual cone of a setOr more generally, if is a nonempty convex cone then the bipolar cone is given by
Relation to the [Fenchel–Moreau theorem]
Letbe the indicator function for a cone
Then the convex conjugate,
is the support function for and
Therefore, if and only if